Is there any web app that people can translate texts (articles, movie subtitles, etc) together?
http://mygengo.com/string/ Might fit the bill, depending on the type of data you need to translate. String is more Web development oriented, but depening on exactly what you want to do, it could be useful
There's a service called Transifex that does the same thing for open source software projects. While it wouldn't work perfectly for an article, I don't see why it cannot be used.
I haven't used it extensively, even less in such a scenario, but it might be worth taking a look. A limitation is that it's for open source and public, which you might not want in all cases.
Crowdin lets teams do translations in a social way, but it is more for software-related projects.
I don't think this is quite what you're looking for, but the premise is strongly displayed here as a social community. From that link:
XIHA (www.xihalife.com) connects the people of the world through a multilingual, cross-cultural exchange. Simultaneous, real-time translation allows XIHA citizens to share travel tips, pictures, music, videos, games, knowledge and perspectives.
By transcending language barriers, XIHA allows users to make new friends from around the globe, rather than merely reconnecting with high school acquaintances. Users can create blogs and homepages - or link to other social media- to share their personal experiences.
Here is a good post talking about the social translation project that TED is running to translate their videos into other languages.
One of the tools they refer to, CONYAC (conyac.cc), might be what you are looking for.
There are also several other good links on Ethan's blog if you read through it.
Do you require any special tools for translation, or just basic text editing?
If basic collaborative text editing is sufficient, you can use WebClipper, which allows you to annotate any web page collaboratively.
Cucumis is a community of translators who share their linguistic knowledge and help each other online.
You can run your own app using:
Pootle - http://translate.sourceforge.net/wiki/. Some wordpress translations are beeing done using this tool.
Try www.wikitranslation.org . You submit texts for translation and the online community can translate it for you.